Generalized $\alpha$-variation and Lebesgue equivalence to differentiable functions
Volume 205 / 2009
Abstract
We find conditions on a real function $f:[a,b]\to\mathbb R$ equivalent to being Lebesgue equivalent to an $n$-times differentiable function ($n\geq 2$); a simple solution in the case $n=2$ appeared in an earlier paper. For that purpose, we introduce the notions of $CBVG_{1/n}$ and $SBVG_{1/n}$ functions, which play analogous rôles for the $n$th order differentiability to the classical notion of a $VBG_*$ function for the first order differentiability, and the classes $CBV_{1/n}$ and $SBV_{{1}/{n}}$ (introduced by Preiss and Laczkovich) for $C^n$ smoothness. As a consequence, we deduce that Lebesgue equivalence to an $n$-times differentiable function is the same as Lebesgue equivalence to a function $f$ which is $(n-1)$-times differentiable with $f^{(n-1)}(\cdot)$ pointwise Lipschitz. We also characterize functions that are Lebesgue equivalent to $n$-times differentiable functions with a.e. nonzero derivatives. As a corollary, we establish a generalization of Zahorski's Lemma for higher order differentiability.
